Linear Algebra, 2020a

384 Chapter Four. Determinants
This picture and definition clarifies central projection but there is still
something ungainly about the dome model: what happens when P looks down?
Consider, in the sketch above, the part of P’s line of sight that comes up towards
us, out of the page. Imagine that this part of the line falls, to the equator and
below. Now the part of the line l that intersects the dome lies behind the page.
That is, as the line of sight continues down past the equator, the projective
point suddenly shifts from the front of the dome to the back of the dome. (This
brings out that the dome does not include the entire equator or else when the
viewer is looking exactly along the equator then there would be two points in the
line that are both on the dome. Instead we define the dome so that it includes
the points on the equator with a positive y coordinate, as well as the point
where y = 0 and x is positive.) This discontinuity means that we often have to
treat equatorial points as a separate case. So while the railroad track model of
central projection has three cases, the dome has two.
We can do better, we can reduce to a model having a single case. Consider a
sphere centered at the origin. Any line through the origin intersects the sphere
in two spots, said to be antipodal. Because we associate each line through
the origin with a point in the projective plane, we can draw such a point as a
pair of antipodal spots on the sphere. Below, we show the two antipodal spots
connected by a dotted line to emphasize that they are not two different points,
the pair of spots together make one projective point.
While drawing a point as a pair of antipodal spots on the sphere is not as intuitive
as the one-spot-per-point dome mode, on the other hand the awkwardness of
the dome model is gone in that as a line of view slides from north to south, no
sudden changes happen. This central projection model is uniform.
So far we have described points in projective geometry. What about lines?
What a viewer P at the origin sees as a line is shown below as a great circle, the
intersection of the model sphere with a plane through the origin.